.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/Interactive/plot_kikuchi_sphere.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_Interactive_plot_kikuchi_sphere.py: Turning a crystal — Kikuchi sphere and detector =============================================== In electron backscatter diffraction (EBSD) every lattice plane of a crystal scatters electrons into a band on a sphere around the sample — a *Kikuchi band*, as wide as twice the plane's Bragg angle. The bands are fixed to the crystal, so the sphere turns with it. A flat detector, fixed in the microscope, only sees the patch of that sphere facing it; turn the crystal and a different patch, with a different set of bands, lands on the detector. * **Left** — the band sphere of nickel (face-centred cubic) at 20 kV, wrapped onto a sphere with :meth:`~anyplotlib.Plot3D.set_texture`. :meth:`~anyplotlib.Plot3D.set_drag_mode` ``("rotate")`` makes a drag turn the *crystal* under a fixed camera, rather than walk the camera around it: drag to roll it, **Shift**-drag to spin it about the line of sight. Two :meth:`~anyplotlib.Plot3D.add_patch` overlays show the two frames a patch can live in: the pink outline is the detector's footprint, fixed in the microscope (``frame="world"``) while the crystal turns under it; the yellow trace is the (111) plane, fixed to the crystal (``frame="data"``) and turning with it. * **Right** — what a detector facing you records: the same sphere projected from its centre onto a flat screen (a gnomonic projection), recomputed in Python from :attr:`~anyplotlib.Plot3D.rotation` on every drag frame. The camera's azimuth and elevation can only aim it, which is two degrees of freedom. An orientation has three, which is why the crystal's orientation is a separate rotation matrix (:meth:`~anyplotlib.Plot3D.set_rotation`) that the camera then looks at. The band model is kinematic and deliberately small — each plane family draws a flat band weighted by a falling atomic scattering factor — so it shows where the bands are, not their real intensity profiles. .. GENERATED FROM PYTHON SOURCE LINES 35-211 .. raw:: html

.. raw:: html .. code-block:: Python import itertools import numpy as np import anyplotlib as apl # ── The crystal: nickel, fcc, a = 0.3524 nm, at 20 kV ─────────────────────── LATTICE_NM = 0.3524 VOLTAGE_V = 20e3 def electron_wavelength_nm(volts): """Relativistic electron wavelength.""" h, m0, e, c = 6.62607e-34, 9.10938e-31, 1.602177e-19, 2.997925e8 momentum = np.sqrt(2 * m0 * e * volts * (1 + e * volts / (2 * m0 * c**2))) return h / momentum * 1e9 WAVELENGTH_NM = electron_wavelength_nm(VOLTAGE_V) def fcc_planes(max_index=3): """One normal per band — ±(hkl) is the same plane — for the families an fcc lattice reflects: h, k, l all odd or all even.""" planes = [] for hkl in itertools.product(range(-max_index, max_index + 1), repeat=3): h, k, l = hkl if hkl == (0, 0, 0) or len({h % 2, k % 2, l % 2}) != 1: continue first_nonzero = next(v for v in hkl if v != 0) if first_nonzero < 0: # keep one of ±(hkl) continue planes.append(hkl) return np.array(planes, dtype=float) PLANES = fcc_planes() G_LENGTH = np.linalg.norm(PLANES, axis=1) / LATTICE_NM # nm⁻¹ BRAGG = np.arcsin(WAVELENGTH_NM * G_LENGTH / 2) # rad WEIGHT = np.exp(-0.12 * np.sum(PLANES**2, axis=1)) # scattering falls with |g| NORMALS = PLANES / np.linalg.norm(PLANES, axis=1, keepdims=True) def band_intensity(directions): """Kinematic band intensity for unit *directions* (…, 3) in the crystal frame: inside a band means within its Bragg angle of the plane. Bragg angles are a degree or two, so the sine of the angle from the plane (a dot product) stands in for the angle itself, and the band edge is a short linear ramp. Done in blocks so the (directions × planes) table stays a few MB. """ flat = directions.reshape(-1, 3).astype(np.float32) normals = NORMALS.astype(np.float32) half_width = np.sin(BRAGG).astype(np.float32) edge = (0.3 * half_width).astype(np.float32) weight = WEIGHT.astype(np.float32) out = np.empty(len(flat), np.float32) for start in range(0, len(flat), 32768): sine_from_plane = np.abs(flat[start:start + 32768] @ normals.T) inside = np.clip((half_width - sine_from_plane) / edge + 0.5, 0, 1) out[start:start + 32768] = inside @ weight return out.reshape(directions.shape[:-1]) def to_grey(values): low, high = np.percentile(values, [1, 99.5]) return np.clip((values - low) / (high - low), 0, 1) # ── The sphere: longitude along the columns, latitude down the rows ───────── # The texture is sampled on a finer copy of the same grid, so the image lines # up with set_texture's default parametric mapping. def sphere_grid(rows, cols): longitude = np.linspace(0, 2 * np.pi, cols) latitude = np.linspace(np.pi / 2, -np.pi / 2, rows) LON, LAT = np.meshgrid(longitude, latitude) return np.stack([np.cos(LAT) * np.cos(LON), np.cos(LAT) * np.sin(LON), np.sin(LAT)], axis=-1) grey = (to_grey(band_intensity(sphere_grid(512, 1024))) * 255).astype(np.uint8) texture = np.repeat(grey[..., None], 3, axis=-1) # set_texture wants RGB mesh = sphere_grid(49, 97) # coarse: the texture carries detail # ── The detector: a flat screen facing the camera ─────────────────────────── # At azimuth 0 and elevation 0 the camera looks along +y, so the face of the # sphere toward it is −y, with +x to the right and +z up on screen. The # detector sits on that side; the pattern centre is straight ahead. DETECTOR_PX = 200 HALF_FIELD = np.tan(np.radians(48)) # ±48° from the pattern centre u = np.linspace(-HALF_FIELD, HALF_FIELD, DETECTOR_PX) U, W = np.meshgrid(u, u[::-1]) # row 0 is the top of the screen rays = np.stack([U, -np.ones_like(U), W], axis=-1) rays /= np.linalg.norm(rays, axis=-1, keepdims=True) # microscope frame def detector_pattern(rotation): """What the detector records with the crystal turned by *rotation*. *rotation* maps crystal directions into the microscope frame, so a ray arriving at the detector came from crystal direction ``rotationᵀ · ray``. """ return to_grey(band_intensity(rays @ rotation)) def facing_direction(rotation): """The crystal direction at the pattern centre, as a short [uvw].""" d = rotation.T @ np.array([0.0, -1.0, 0.0]) return "[" + " ".join(f"{v:+.2f}" for v in d) + "]" def bunge(phi1, Phi, phi2): """Rotation from Bunge Euler angles in degrees (z-x-z).""" a, b, c = np.radians([phi1, Phi, phi2]) def rz(t): return np.array([[np.cos(t), -np.sin(t), 0], [np.sin(t), np.cos(t), 0], [0, 0, 1]]) rx = np.array([[1, 0, 0], [0, np.cos(b), -np.sin(b)], [0, np.sin(b), np.cos(b)]]) return rz(a) @ rx @ rz(c) start = bunge(20, 40, 15) # ── Figure ────────────────────────────────────────────────────────────────── fig, (ax_sphere, ax_detector) = apl.subplots( 1, 2, figsize=(900, 450), help="Drag the sphere to turn the crystal; Shift-drag spins it about\n" "the line of sight. The detector shows the patch facing you.") sphere = ax_sphere.plot_surface(mesh[..., 0], mesh[..., 1], mesh[..., 2], bounds=((-1, 1),) * 3, azimuth=0, elevation=0, zoom=1.35) sphere.set_axis_off() sphere.set_texture(texture, shade=True, cull_backfaces=True) sphere.set_rotation(start) sphere.set_drag_mode("rotate") sphere.set_title("Kikuchi sphere · Ni, 20 kV") # The detector's footprint: its four corners, seen from the sphere's centre. # A flat detector's straight edges project onto great circles, which is what # add_patch draws between corners by default. corners = np.array([[sx * HALF_FIELD, -1.0, sz * HALF_FIELD] for sx, sz in ((-1, -1), (1, -1), (1, 1), (-1, 1))]) corners /= np.linalg.norm(corners, axis=1, keepdims=True) sphere.add_patch(corners, frame="world", facecolor="#f38ba8", alpha=0.12, linewidth=2, label="detector") # The (111) plane's trace: a great circle through four points 90° apart. normal = np.ones(3) / np.sqrt(3) in_plane = np.cross(normal, [0.0, 0.0, 1.0]) in_plane /= np.linalg.norm(in_plane) across = np.cross(normal, in_plane) sphere.add_patch([in_plane, across, -in_plane, -across], frame="data", edgecolor="#f9e2af", alpha=0, linewidth=1.5, label="(111)") detector = ax_detector.imshow(detector_pattern(start), cmap="gray", vmin=0, vmax=1) detector.set_title(f"Detector · centre {facing_direction(start)}") @sphere.add_event_handler("pointer_move") def follow_the_crystal(event): # Every drag frame: ~40k rays against ~50 bands is a few milliseconds. if event.rotation is None: # a wheel zoom, not a turn return rotation = sphere.rotation detector.set_data(detector_pattern(rotation), clim=(0, 1)) detector.set_title(f"Detector · centre {facing_direction(rotation)}") fig # Interactive .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.017 seconds) .. _sphx_glr_download_auto_examples_Interactive_plot_kikuchi_sphere.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_kikuchi_sphere.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_kikuchi_sphere.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_kikuchi_sphere.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_